Senior High School
Statistics and Probability
Second Semester– Module 1.1
Random Variable
SLM
SELF-LEARNING MODULE
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Senior High School
Statistics and Probability
Second Semester – Module 1.1
Random Variable
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Introductory Message
For the facilitator:
Welcome to the Basic Calculus Self-Learning Module on Random Variable!
This module was collaboratively designed, developed and reviewed by educators
both from public and private institutions to assist you, the teacher or facilitator in helping
the learners meet the standards set by the K to 12 Curriculum while overcoming their
personal, social, and economic constraints in schooling.
This learning resource hopes to engage the learners into guided and independent
learning activities at their own pace and time. Furthermore, this also aims to help learners
acquire the needed 21st century skills while taking into consideration their needs and
circumstances.
In addition to the material in the main text, you will also see this box in the body of
the module:
Notes to the Teacher
This contains helpful tips or strategies that will help
you in guiding the learners.
As a facilitator, you are expected to orient the learners on how to use this module. You
also need to keep track of the learners' progress while allowing them to manage their
own learning. Furthermore, you are expected to encourage and assist the learners as
they do the tasks included in the module.
For the learner:
Welcome to the Basic Calculus Self-Learning Module on Random Variable!
This module was designed to provide you with fun and meaningful opportunities for
guided and independent learning at your own pace and time. You will be enabled to
process the contents of the learning resource while being an active learner.
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Random Variable
CONTENT STANDARD
The learner demonstrates understanding of key concepts of random variables and
probability distributions.
PERFORMANCE STANDARD
The learner is able to apply an appropriate random variable for a given real-life
problem (such as in decision making and games of chance).
LEARNING COMPETENCY
Illustrates a random variable (discrete and continuous). MII/12SP-IIIa-1
LEARNING OBJECTIVES:
1. Illustrate random variables.
2. Give the possible outcomes of a random variable.
3. Relate the concept to real-life problems.
INTRODUCTION
Most of the improvements the world is enjoying today are attributed to the
understanding of the nature of information or data that occurred in any experimental
event. These are the characteristics of certain variable randomly observed as an
outcome of an experiment of chance. This section set forth to bring about understanding
the concepts, illustrating a random variable
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PRE-TEST
Name: _______________________ Grade Level & Section: __________ Score: ______
Read the questions carefully. Choose the letter that corresponds to the correct answer.
Write the correct answer in the blank that corresponds to your answer.
_____ 1. Which of the following is a discrete random variable?
A. the number of patients in a hospital
B. the average weight of female athletes
C. the average amount of electricity consumed
D. the amount of paint used in repainting a building
_____ 2. If two coins are tossed, which is NOT a possible value of the random variable
for the number of heads?
A. 0
B. 1
C. 2
D. 3
_____ 3. Which of the following is NOT a true statement?
A. Random variables can only have one value.
B. The value of a random variable could be zero.
C. The probability of the value of a random variable could be zero.
D. The sum of all the probabilities in a probability distribution is always
equal to one.
_____ 4. Find the probability of drawing the King of Spades from a complete deck of
cards.
A. 1/52
B. 26/52
C. 30/52
D. 52/52
_____ 5. How many outcomes are there if a coin is tossed?
A. 1
B. 2
C. 3
D. 4
_____ 6. Find the probability of getting an even number in the roll of a die.
A. ½
B. 1/3
C. 2/3
D. 5/6
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_____ 7. In rolling two dice, how many outcomes are possible?
A. 6
B. 36
C. 120
D. none of these
_____ 8. In the experiment of tossing a fair coin 3 times, the probability distribution of
showing the number of tails may be observed. What is the probability of getting 3 tailed?
A. 0.125
B. 0.250
C. 0.375
D. none of these
_____ 9. Find the probability of getting 1 in the roll of a die
A. 1/2
B. 1/6
C. 2/3
D. 5/6
____10. Find the probability of obtaining 2 heads at the toss of a coin twice.
A. 1
B. ½
C. ¼
D. 0
____11. The set of all possible outcomes of an experiment is called
A. data
B. population
C. sample space
D. statistics
____12. It is the characteristic or property of the element.
A. continuous variable
B. discrete variable
C. random variable
D. variable
____13. Which of the following examples is a continuous random variable?
A. the height of each student in a class
B. the number of people in each country
C. the number of siblings of your classmates
D. the number of correct answers in a 5-item true or false quiz
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____14. Identify which of the following qualifies as a discrete random variable.
A. body weight
B. number of siblings
C. shoe size
D. water consumption
____15. In an experiment of tossing three coins, how many elements are there in the
sample space?
A. 4
B. 5
C. 6
D. 8
REVIEW OF THE PREVIOUS MODULE
You have learned in your past lessons that Statistics is a branch of mathematics
dealing with the collection, analysis, interpretation and presentation of masses of
numerical data. You also learned that in mathematics an experiment is any activity, which
can be done repeatedly under similar conditions. The set of all possible outcomes of an
experiment is called the sample space.
PRESENTATION OF THE NEW MODULE
Basketball is a very famous sport in the country nowadays. Professional basketball
teams scout players from the amateur basketball leagues on either colleges/universities
or local communities. Basketball players are being accepted or draft pick based on their
performance in the game. If you are the manager of a basketball team, how would you
answer the following inquiries?
1. What information should be obtained to select the players your team
needs?
2.
How do you count or measure the information needed for making decision?
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ACTIVITY
Part 1. Identifying Domain and Range of a Function
• {(2,5),3,7),(4,9),(5,11)(6,13),(7,15),(8,17),(9,19)}
Write first all the elements of the domain and then all the elements of the range.
The elements of the domain are: ___________________________________________
The elements of the range are: ____________________________________________
Part 2. Identifying the Elements of a Set
• Consider the set of colors in a rainbow.
Write the different colors that you have seen in the rainbow
The
different
colors
of
the
rainbow
are:
______________________
_____________________________________________________________________
ANALYSIS
For Part 1
•
What is the domain of the set?
•
How about range of the set?
For Part 2
•
Are all of those elements constitute the sample space of that set?
•
Can you name a set of things that could be seen inside your room?
•
What is a set?
•
What is the domain of the set?
•
What is the range of the set?
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ABSTRACTION
Definitions:
Variable – is the characteristic or property of the element.
Random variable
• is a variable being measured to produce numerical observations associated
with the random outcomes of a chance experiment.
• is a function whose domain is the sample space of a random experiment, and
the range of the random variable is the smallest interval that contains all the
values of the random variable.
• it is typically represented by an uppercase letter, usually X, while its
corresponding lowercase letter in this case, x, is used to represent one of its
values.
Sample space – is the set of all possible outcomes of an experiment.
Event – outcomes of an experiment.
Example 1:
Consider a random experiment of tossing a fair coin three times. In this scenario,
the domain can be defined as the set of all possible outcomes of the experiment and the
range of the random variable as the total number of tails that comes out after tossing a
coin three times.
Let X (the random variable).- be the number of heads in the tossing of fair coins
three times.
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1st toss
2nd toss
3rd toss
H
H
T
H
T
T
Possible
Outcomes
H
HHH
T
HHT
H
HTH
T
HTT
H
THH
T
THT
H
TTH
T
TTT
For each element in the domain, there is a corresponding value for the random
variable X. The specific value of a random variable is denoted by small letter x. The
domain and range of the random variable X are shown in the table below:
Value of the Random Variable X
(Numbers of Heads)
HHH
3
HHT
2
HTH
2
THH
2
THT
1
TTH
1
HTT
1
TTT
0
In the illustration above, random variable is represented by the upper-case X. The
lower-case x represents the specific values. Hence, x = 3, x = 2, x = 2, x = 2, x = 1, x =
1, x = 1, and x = 0.
The sample space or the set of possible outcomes (domain) of the experiment is as
follows:
Possible Outcomes
S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}.
The value of the variable X can be 0, 1, 2, or 3. Then, in this example, X is a
random variable.
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A random variable X is a variable whose value is determined by chance. We are
dealing with an entity that varies in value and whose behavior is governed by chance. If
X is the random variable then x is the observed numerical value of X. A random variable
does not have a fixed, single value rather takes on different values, and for each value
there is an associated probability. The following examples provide a collection of random
variables for illustrative purposes.
Examples:
1. An experiment consists of rolling a pair of dice. The random variable x is
defined by X = sum of the spots on the two dice. Then one possible value for
X is x = 7. This is the event when the sum of two dice equals 7.
2. The morning shift at the fire station begins at 6:00 a.m. and past experience
indicates that at least one call will be logged by 8:00 a.m. The random
variable T denotes the time at which the first call of the day is received.
Each of the above examples has also associated with it a random variable. Each
variable X, T, can change in value from trial to trial, and chance is certainly a factor.
Before an experiment, we cannot say exactly what the outcome will be, after each
experiment, the variable will have assumed a numerical value determined by chance.
There are two types of random variables:
(1) Discrete random variable
(2) Continuous random variable.
Discrete random variable – a random variable that can assume at most a finite or
a countable number of possible values.
Example: The random variable X is the sum of the spots on two dice is discrete
since x can assume only the values 2,3,4,5,6,7,8,9,10,11 and 12. The number of possible
outcomes is finite or can be counted.
Continuous random variable – is one that can assume any value in some interval or
intervals of real numbers, and the probability that it assumes any specific value is greater
than zero. This random variable has no gaps.
Example: The random variable T is the time that the first call of the day is received
at the fire station is continuous. If we attempted to list possibilities, we find it impossible
to do so. We might list some outcomes with an interval of 10 minutes like 6:10, 6:20,
6:30, 6:40. The list is clearly incomplete since the first call might arrive at 6:05. Thus, we
pose the following possibilities: 6:00, 6:05, 6:10, 6:15, 6:20. This is an improvement but
still incomplete. Between any two real numbers, there exists another real number that is
possible value of T, time being measured continuously. Therefore, to describe the set of
values for T, it would be in the interval of 6:00 and 8:30.
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APPLICATION
A. Four coins are tossed. Let X be the random variable representing the number of
heads that occur. Find the values of the random variable X.
Value of the Random Variable X
(Numbers of Heads)
Possible Outcomes
ENRICHMENT
Suppose an experiment is conducted to determine the distance that a certain type
of car will travel using 10 liters of gasoline over a prescribed test course. If distance is a
random variable, then we have an infinite number of distances that cannot be equated to
the number of whole numbers. This is an example of a continuous random variable.
1.
2.
3.
4.
How do you describe a discrete random variable?
How do you describe a continuous random variable.
Give three examples of discrete random variable.
Give three examples of continuous random variable.
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REFERENCES
•
Teaching Guide for Senior High School Statistics & Probability
•
Statistics and Probability Book
•
Statistics and Probability by Rene R. Belecina, et.al, Rex Book Store
•
Statistics and Probability Vibal by Calaca, Uy, Nobl, Manalo,
•
Elementary Statistics with Computer Applications by Priscilla S. Altares, et.al,
Second Edition Rex Book Store
•
•
Next Century Mathematics Statistics & Probability by Jesus P. Mercado, et.al,
Phoenix Publishing House
•
www.quipperschool.com,
•
https://www.onlinemathlearning.com/random-variable.html
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For inquiries or feedback, please write or call:
Department of Education – Schools Division of Surigao del Norte
Peñaranda St., Surigao City
Surigao del Norte, Philippines 8400
Tel. No: (086) 826-8216
Email Address: surigao.delnorte@deped.gov.ph
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